Journal of Computational and Applied Mathematics
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This part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in $\mathbb R^N$. Global estimates were considered in Part I. In the sense to be discussed below, these sharpen known interior quasi-optimal $L_\infty$ and $W^1_\infty$ estimates in that they indicate a more local dependence of the error at a point on the derivatives of the solution near the point. The higher the order of the finite element the more local the behavior of the finite element approximation. As a consequence of these estimates, new types of local error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in $\mathbb R^N$ and have applications to superconvergence, extrapolation, and a posteriori estimates for both smooth and nonsmooth problems.